0
TECHNICAL BRIEFS

Stochastic Analysis of a 1-D System With Fractional Damping of Order 1/2

[+] Author and Article Information
Om P. Agrawal

Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901e-mail: om@engr.siu.edu

J. Vib. Acoust 124(3), 454-460 (Jun 12, 2002) (7 pages) doi:10.1115/1.1471357 History: Received March 01, 1999; Revised February 01, 2002; Online June 12, 2002
Copyright © 2002 by ASME
Your Session has timed out. Please sign back in to continue.

References

Bagley,  R. L., and Torvik,  P. J., 1983, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” J. Rheol., 27, pp. 201–210.
Bagley,  R. L., and Torvik,  P. J., 1983, “Fractional Calculus—A Different Approach to the Analysis of Viscoelastically Damped Structures,” AIAA J., 21, pp. 741–748.
Bagley,  R. L., and Torvik,  P. J., 1985, “Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures,” AIAA J., 23, pp. 918–925.
Koeller,  R. C., 1984, “Application of Fractional Calculus to the Theory of Viscoelasticity,” ASME J. Appl. Mech., 51, pp. 299–307.
Makris,  N., and Constantinou,  M. C., 1991, “Fractional-Derivative Maxwell Model for Viscous Dampers,” Journal of Structural Engineering, 117, pp. 2708–2724.
Mainardi,  F., 1994, “Fractional Relaxation in Anelastic Solids,” J. Alloys Compd., 211-1, pp. 534–538.
Shen,  K. L., and Soong,  T. T., 1995, “Modeling of Viscoelastic Dampers for Structural Applications,” J. Eng. Mech., 121, pp. 694–701.
Pritz,  T., 1996, “Analysis of Four-Parameter Fractional Derivative Model of Real Solid Materials,” J. Sound Vib., 195, pp. 103–115.
Papoulia,  K. D., and Kelly,  J. M., 1997, “Visco-Hyperelastic Model for Filled Rubbers Used in Vibration Isolation,” ASME J. Eng. Mater. Technol., 119, pp. 292–297.
Makris,  N., and Constantinou,  M. C., 1992, “Spring-Viscous Damper Systems for Combined Seismic and Vibration Isolation,” Earthquake Eng. Struct. Dyn., 21, pp. 649–664.
Lee,  H. H., and Tsai,  C. S., 1994, “Analytical Model for Viscoelastic Dampers for Seismic Mitigation of Structures,” Comput. Struct., 50, No. 1, pp. 111–121.
Skaar,  S. B., Michel,  A. N., and Miller,  R. K., 1988, “Stability of Viscoelastic Control Systems,” IEEE Trans. Autom. Control, 33, pp. 348–357.
Makroglou,  A., Miller,  R. K., and Skaar,  S., 1994, “Computational Results for a Feedback Control for a Rotating Viscoelastic Beam,” J. Guid. Control Dyn., 17, pp. 84–90.
Bagley,  R. L., and Calico,  R. A., 1991, “Fractional Order State Equations for the Control of Viscoelastically Damped Structures,” J. Guid. Control Dyn., 14, pp. 304–311.
Mbodje, B., Montseny, C., Audounet, J., and Benchimol, P., 1994, “Optimal Control for Fractionally Damped Flexible Systems,” The Proceedings of the Third IEEE Conference on Control Applications, The University of Strathclyde, Glasgow, August 24–26, pp. 1329–1333.
Makris,  N., Dargush,  G. F., and Constantinou,  M. C., 1993, “Dynamic Analysis of Generalized Viscoelastic Fluids,” J. Eng. Mech., 119, pp. 1663–1679.
Suarez,  L. E., and Shokooh,  A., 1997, “An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives,” ASME J. Appl. Mech., 64, pp. 629–635.
Rossikhin,  Y. A., and Shitikova,  M. V., 1997, “Application of Fractional Operators to the Analysis of Damped Vibrations of Viscoelastic Single-Mass Systems,” J. Sound Vib., 199, pp. 567–586.
Gaul,  L., Klein,  P., and Kemple,  S., 1989, “Impulse Response Function of an Oscillator with Fractional Derivative in Damping Description,” Mech. Res. Commun., 16, pp. 297–305.
Gaul,  L., Klein,  P., and Kemple,  S., 1991, “Damping Description Involving Fractional Operators,” Mech. Syst. Signal Process., 5, pp. 8–88.
Padovan,  J., 1987, “Computational Algorithms for Finite Element Formulation Involving Fractional Operators,” Computational Mechanics, 2, pp. 271–287.
Koh,  C. G., and Kelly,  J. M., 1990, “Application of Fractional Derivatives to Seismic Analysis of Base-Isolated Models,” Earthquake Eng. Struct. Dyn., 19, pp. 229–241.
Lixia, Y., and Agrawal, O. P., 1998, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives,” ASME J. Vibr. Acoust., in press.
Riewe,  F., 1996, “Nonconservative Lagrangian and Hamiltonian mechanics,” Phys. Rev. E , 53, pp. 1890–1899.
Riewe,  F., 1997, “Mechanics with Fractional Derivative,” Phys. Rev. E, 55, pp. 3581–3592.
Mainardi, F., 1997, “Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics,” Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, eds., Springer Verlag, Wien, New York, pp. 291–348.
Spanos,  P. D., and Zeldin,  B. A., 1997, “Random Vibration of Systems with Frequency-Dependent Parameters or Fractional Derivatives,” J. Eng. Mech., 123, pp. 290–292.
Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic Press, New York, NY.
Miller and Ross, 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, NY.
Samko, S. G., Kilbas, A. A., and Marichev, O. I., 1993, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach Science Publishers, Longhorne, PA.
Kiryakova, V. S., 1993, Generalized Fractional Calculus and Applications, Longman Scientific & Technical, Longman House, Burnt Mill, Harlow, England.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, New York, NY.
Gorenflo, R., and Mainardi, F., 1997, “Fractional Calculus: Integral and Differential Equations of Fractional Order,” Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, eds., Springer Verlag, Wien, New York, pp. 223–276.
Butzer P. L., and Westphal, U, 2000, “An Introduction to Fractional Calculus,” Applications of Fractional Calculus in Physics, R. Hilfer, ed., World Scientific, New Jersey, pp. 1–85.
Spiegel, M. R., 1998, Advanced Mathematics for Engineers and Scientists, McGraw Hill, New York, NY.
Lin, Y. K., 1965, Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, NY.
Nigam, N. C., 1983, Introduction to Random Vibrations, MIT Press, Cambridge, MA.

Figures

Grahic Jump Location
Variance function E[x2] as a function of time (under damped condition)
Grahic Jump Location
Variance function E[ν2] as a function of time (under damped condition)
Grahic Jump Location
Covariance function E[xν] as a function of time (under damped condition)
Grahic Jump Location
Variance function E[x2] as a function of time (under, critical, and over damped conditions)
Grahic Jump Location
Variance function E[ν2] as a function of time (under, critical, and over damped conditions)
Grahic Jump Location
Covariance function E[xν] as a function of time (under, critical, and over damped conditions)
Grahic Jump Location
Variance of E[x2] for fractional model and Models 1 and 2 (η=0.05)
Grahic Jump Location
Variance of E[ν2] for fractional model and Models 1 and 2 (η=0.05)
Grahic Jump Location
Covariance of E[xν] for fractional model and Models 1 and 2 (η=0.05)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In